BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

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Add DISPLAYTITLE: title the page by its section rather than its path

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			{{DISPLAYTITLE:Part II — Prop. VI}}
	''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part II\|Part II]] → Prop. VI''		''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part II\|Part II]] → Prop. VI''

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	''Next: [[On Physical Lines of Force/Part II/Prop. VII\|Prop. VII]] · Previous: [[On Physical Lines of Force/Part II/Prop. V\|Prop. V]]''		''Next: [[On Physical Lines of Force/Part II/Prop. VII\|Prop. VII]] · Previous: [[On Physical Lines of Force/Part II/Prop. V\|Prop. V]]''

BFP Rauch: 1 revision imported: Maxwell, On Physical Lines of Force — initial import

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1 revision imported: Maxwell, On Physical Lines of Force — initial import

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Imported from the DokuWiki edition

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''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part II|Part II]] → Prop. VI''

== Maxwell's statement ==

<blockquote>
'''Prop. VI.''' — To determine the actual energy of a portion of a medium due to the motion of the vortices within it.
</blockquote>

"Actual energy" is the term of the period for what we now call kinetic energy, as opposed to "potential energy". Maxwell is asking: how much energy is stored in the spinning?

== The argument ==

The energy of a rotating body goes as its mass times the square of its speed. In the medium, the mass is carried by the density <math>\mu</math> and the speed is the circumferential velocity, whose components are <math>\alpha,\beta,\gamma</math>. So the energy of a volume <math>V</math> must have the form

<math display="block">E \;=\; C\,\mu\,(\alpha^{2}+\beta^{2}+\gamma^{2})\,V</math>

with <math>C</math> an unknown constant that depends on how the angular velocity and density are distributed inside each vortex — the same unspecified distribution that [[On Physical Lines of Force/Part I/Prop. I|Prop. I]] left free.

Maxwell fixes <math>C</math> by conservation of energy, and does it with an argument of real elegance. Move one magnetic system relative to another. The mechanical work done is already known: it is given by equations '''(12)''' of [[On Physical Lines of Force/Part I/Prop. III|Part I]], the force per unit volume. And that work must come from somewhere — it comes out of the energy stored in the vortices. Equating the two determines <math>C</math>.

The result is equation '''(46)''':

<math display="block">E \;=\; \frac{\mu}{8\pi}\,(\alpha^{2}+\beta^{2}+\gamma^{2})\;V \qquad\qquad\mathrm{(46)}</math>

''Provenance: the equation number (46) is attested — Maxwell cites it by number in Prop. VIII ("Let V be the volume of a vortex, then by (46) its energy is …"). The body is a reconstruction; the coefficient <math>1/8\pi</math> is the value required by consistency with equations (12) and is the standard result. See [[On Physical Lines of Force/Sources and editorial method|Sources]].''

== What this equation is ==

Divide by the volume and it reads: the energy stored per unit volume of the magnetic field is

<math display="block">u \;=\; \frac{\mu H^{2}}{8\pi} \;=\; \frac{\mathbf{B}\cdot\mathbf{H}}{8\pi}</math>

which is '''the magnetic energy density''', still written that way in Gaussian units today. It enters physics here, as the kinetic energy of an invisible flywheel.

That reading is not a retrospective gloss. It is the whole point of the proposition, and Maxwell uses it as such immediately: in [[On Physical Lines of Force/Part II/Prop. VIII|Prop. VIII]] he equates the rate of change of this energy with the work supplied by the particles, and Faraday's law falls out. The energy of the magnetic field is treated from the start as ''mechanical'' energy of an actual moving medium, with all the consequences that carries — it can be stored, it can be given back, and its rate of change must be accounted for.

== Why the mass of the particles does not appear ==

Because Maxwell said in [[On Physical Lines of Force/Part II/Prop. V|Prop. V]] that

<blockquote>
the mass of all the particles together is inappreciable compared with that of the vortices
</blockquote>

so all the kinetic energy sits in the rotation and none in the translation of the electricity. This is a real physical commitment, and it is what makes the model's inductance purely magnetic. A modern reader may note that the electron's mass does contribute in reality, and that Maxwell's assumption is the reason his theory has no analogue of the inertia of charge carriers.

== The bridge to induction ==

Once you have

* an expression for the energy stored in the field (Prop. VI), and
* an expression for the rate at which the particles do work on the vortices ([[On Physical Lines of Force/Part II/Prop. VII|Prop. VII]]),

then equating the two is a pure piece of energy bookkeeping, and it must produce a relation between the rate of change of the magnetic field and the forces on the electricity. That relation is Faraday's law. [[On Physical Lines of Force/Part II/Prop. VIII|Prop. VIII]] does exactly this.

This is the structural insight of Part II, and it is worth stating flatly: '''Maxwell derives electromagnetic induction from conservation of energy in a rotating mechanical system.''' He does not postulate it. Faraday's law arrives as a bookkeeping identity.

----

''Next: [[On Physical Lines of Force/Part II/Prop. VII|Prop. VII]] · Previous: [[On Physical Lines of Force/Part II/Prop. V|Prop. V]]''

On Physical Lines of Force/Part II/Prop. VI - Revision history

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

BFP Rauch: 1 revision imported: Maxwell, On Physical Lines of Force — initial import

opl>Import: Imported from the DokuWiki edition